Question: Below is the graph of $y = a \tan bx$ for some positive constants $a$ and $b.$  Find $ab.$

[asy]import TrigMacros;

size(250);

real g(real x)
{
	return 2*tan(3/2*x);
}

draw(graph(g,-pi + 0.01,-pi/3 - 0.01),red);
draw(graph(g,-pi/3 + 0.01,pi/3 - 0.01),red);
draw(graph(g,pi/3 + 0.01,pi - 0.01),red);
limits((-pi,-4),(pi,4),Crop);
draw((-pi/3,-4)--(-pi/3,4),dashed);
draw((pi/3,-4)--(pi/3,4),dashed);
trig_axes(-pi,pi,-4,4,pi/6,1);
layer();
rm_trig_labels(-5, 5, 6);

label("$1$", (0,1), E);
label("$2$", (0,2), E);
label("$3$", (0,3), E);
label("$-1$", (0,-1), E);
label("$-2$", (0,-2), E);
label("$-3$", (0,-3), E);
[/asy]
The graph has period $\frac{2 \pi}{3}.$  The period of $y = a \tan bx$ is $\frac{\pi}{b},$ so $b = \frac{3}{2}.$

The graph is then of the form
\[y = a \tan \left( \frac{3x}{2} \right).\]Since the graph passes through $\left( \frac{\pi}{6}, 2 \right),$
\[2 = a \tan \frac{\pi}{4} = a.\]Therefore, $ab = 2 \cdot \frac{3}{2} = \boxed{3}.$